# Multi-step ahead forecasting with Weighted Moving Average?

The Weighted Moving Average method is usually used for smoothing purposes. However, it can be used to forecast $Y(t+1)$ based on the last n observed data.

In real-world problems, forecasting in very short horizon $(h = 1)$ is not very interesting.

I would like to know if it is possible to use larger forecasting horizons in WMA method like in HoltWinter and ARIMA techniques ?

• Is it possible?: Sure, why not? The WMA is a recursive formula, so to get an n-step ahead forecast, plug in the previous (n-1)-step ahead forecasts to the right hand side. You can can choose some data or simulate it. See how well the WMA forecast performs for multiple steps out of sample and compare it to other techniques like ARIMA and HW. – Zachary Blumenfeld Aug 1 '15 at 12:15
• But calculating $Y(t+h)$ at observation $t$, requires the calculation of $Y(t+h-1), Y(t+h-2), ..., Y(t+h-n)$. That is the weighted average of the last $n$ observed data, which is not available at observation $t$. – mathematix Aug 1 '15 at 22:03

For an exponentially WMA you would get a flat forecast. My answer is derived from the information at http://people.duke.edu/~rnau/411avg.htm . Say you are using an exponentially Weighted Moving average for a series $\{Y_t\}$. You can write it as $$S_{t}= \alpha Y_{t}+(1-\alpha)S_{t-1}$$ Your smoothed value $S_t$ is the point forecast for the next time period, i.e. $\hat Y_{t+1}=S_t$. Thus $$\hat Y_{t+1}= \alpha Y_{t}+(1-\alpha)\hat Y_{t}$$
The only thing that you can do if you wanted to forecast 2 periods into the future is substitute in $\hat Y_{t+1}$ for $Y_{t+1}$ $$\hat Y_{t+2}= \alpha \hat Y_{t+1}+(1-\alpha)\hat Y_{t+1}=\hat Y_{t+1}$$
and so $\hat Y_{t+1}=\hat Y_{t+2}=\hat Y_{t+3}=\hat Y_{t+4}=...$ and so on.